The Two-Nucleon System in Three Dimensions
J.Golak, W.Glockle, R.Skibinski, H.Witala, D.Rozpedzik, K.Topolnicki, I.Fachruddin, Ch.Elster, A.Nogga
aa r X i v : . [ nu c l - t h ] J a n The Two-Nucleon System in Three Dimensions
J. Golak , W. Gl¨ockle , R. Skibi´nski , H. Wita la , D. Rozp ι edzik ,K. Topolnicki , I. Fachruddin , Ch. Elster , and A. Nogga M. Smoluchowski Institute of Physics,Jagiellonian University, PL-30059 Krak´ow, Poland Institut f¨ur Theoretische Physik II,Ruhr-Universit¨at Bochum, D-44780 Bochum, Germany Departemen Fisika, Universitas Indonesia, Depok 16424, Indonesia Institute of Nuclear and Particle Physics,Department of Physics and Astronomy,Ohio University, Athens, OH 45701, USA and Forschungszentrum J¨ulich, Institut f¨ur Kernphysik (Theorie),Institute for Advanced Simulation and J¨ulich Centerfor Hadron Physics, D-52425 J¨ulich, Germany (Dated: September 14, 2018)
Abstract
A recently developed formulation for treating two- and three-nucleon bound states in a three-dimensional formulation based on spin-momentum operators is extended to nucleon-nucleon scat-tering. Here the nucleon-nucleon t-matrix is represented by six spin-momentum operators accom-panied by six scalar functions of momentum vectors. We present the formulation and providenumerical examples for the deuteron and nucleon-nucleon scattering observables. A comparison toresults from a standard partial wave decomposition establishes the reliability of this new formula-tion.
PACS numbers: 21.45.-v, 21.30.-x, 21.45.Bc . INTRODUCTION A standard way to obtain scattering observables for nucleon-nucleon (NN) scattering isto solve the Schr¨odinger equation either in momentum or coordinate space by taking advan-tage of rotational invariance and introduce a partial wave basis. This is a well establishedprocedure and has at low energies (below the pion production threshold) a clear physicalmeaning. At higher energies the number of partial waves needed to obtain converged resultsincreases, and approaches based on a direct evaluation of the scattering equation in termsof vector variables become more appealing.Especially the experience in three- and four-nucleon calculations [1, 2] shows that thestandard treatment based on a partial wave projected momentum space basis is quite suc-cessful at lower energies, but becomes increasingly more tedious with increasing energy, sinceeach building block requires extended algebra and intricate numerical realizations. On theother hand for a system of three bosons interacting via scalar forces the relative ease withwhich a three-body bound state [3] as well as three-body scattering [4] can be calculatedin the Faddeev scheme when avoiding an angular momentum decomposition altogether hasbeen successfully demonstrated. Thus it is only natural to strive for solving the three nucleon(3N) Faddeev equations in a similar fashion.Recently we proposed a three-dimensional (3D) formulation of the Faddeev equationsfor 3N bound states [10] and 3N scattering [11] in which the spin-momentum operators areevaluated analytically, leaving the Faddeev equations as a finite set of coupled equationsfor scalar functions depending only on vector momenta. One of the basic foundations ofthis formulation rests on the fact that the most general form of the NN interaction canonly depend on six linearly independent spin-momentum operators, which in turn dictatethe form of the NN bound and scattering state. Here we extend the formulation of the NNbound state given in [10] to NN scattering and provide a numerical realization.There have been several approaches of formulating NN scattering without employing apartial wave decomposition. A helicity formulation related to the total NN spin was proposedin [5], which was extended to 3N bound state calculations in [6]. The spectator equationfor relativistic NN scattering has been successfully solved in [7] using a helicity formulation.Aside from NN scattering, 3D formulations for the scattering of pions off nucleons [8] andprotons off light nuclei [9] have recently been successfully carried out.In Section II we introduce the formal structure of our approach starting from the mostgeneral form of the NN potential. We derive the resulting Lippmann-Schwinger equationand show how to extract Wolfenstein parameters and NN scattering observables. Numericalrealizations of our approach that employ a recent chiral next-to-next-leading order (NNLO)NN force [12–14] as well as the standard one-boson-exchange potential Bonn B [15] arepresented in Section III. The scalar functions, which result from the evaluation of the spin-momentum operators and have to be calculated only once are given in Appendices A andB. Finally we conclude in Section IV. The more technical information necessary to performcalculations with the chiral potential is given in Appendix C. In Appendix D the Bonn Bpotential is presented in the form required by our formulation.
II. THE FORMAL STRUCTURE
We start by projecting the NN potential on the NN isospin states | tm t i , with t = 0 , m t =0 being the singlet and t = 1 , m t = − , , m t , h t ′ m ′ t | V | tm t i = δ tt ′ δ m t m ′ t V tm t (2.1)Furthermore, the most general rotational, parity and time reversal invariant form of theoff-shell NN force can be expanded into six scalar spin-momentum operators [17], which wechoose as w ( σ , σ , p ′ , p ) = 1 w ( σ , σ , p ′ , p ) = σ · σ w ( σ , σ , p ′ , p ) = i ( σ + σ ) · ( p × p ′ ) w ( σ , σ , p ′ , p ) = σ · ( p × p ′ ) σ · ( p × p ′ ) w ( σ , σ , p ′ , p ) = σ · ( p ′ + p ) σ · ( p ′ + p ) w ( σ , σ , p ′ , p ) = σ · ( p ′ − p ) σ · ( p ′ − p ) (2.2)Each of these operators is multiplied with scalar functions which depend only on the mo-menta p and p ′ , leading to the most general expansion for any NN potential V tm t ≡ X j =1 v tm t j ( p ′ , p ) w j ( σ , σ , p ′ , p ) (2.3)The property of Eq. (2.1) carries over to the NN t-operator, which fulfills the Lippmann-Schwinger (LS) equation t tm t = V tm t + V tm t G t tm t , (2.4)with G ( z ) = ( z − H ) − being the free resolvent. The t-matrix element has an expansionanalogous to the potential, t tm t ≡ X j =1 t tm t j ( p ′ , p ) w j ( σ , σ , p ′ , p ) (2.5)Inserting Eqs. (2.3) and (2.5) into the LS equation (2.4), operating with w k ( σ , σ , p ′ , p )from the left and performing the trace in the NN spin space leads to X j A kj ( p ′ , p ) t tm t j ( p ′ , p ) = X j A kj ( p ′ , p ) v tm t j ( p ′ , p )+ Z d p ′′ X jj ′ v tm t j ( p ′ , p ′′ ) G ( p ′′ ) t tm t j ′ ( p ′′ , p ) B kjj ′ ( p ′ , p ′′ , p ) . (2.6)The scalar coefficients A kj and B kjj ′ are defined as A kj ( p ′ , p ) ≡ Tr (cid:16) w k ( σ , σ , p ′ , p ) w j ( σ , σ , p ′ , p ) (cid:17) (2.7) B kjj ′ ( p ′ , p ′′ , p ) ≡ Tr (cid:16) w k ( σ , σ , p ′ , p ) w j ( σ , σ , p ′ , p ′′ ) w j ′ ( σ , σ , p ′′ , p ) (cid:17) (2.8)Here all spin dependencies are analytically evaluated, and the coefficients only depend on thevectors p , p ′ , and p ′′ . The explicit expressions for the coefficients are given in Appendix A.3hus we end up with a set of six coupled equations for the scalar functions t tm t j ( p ′ , p ),which depend for fixed | p | on two other variables, | p ′ | and the cosine of the relative anglebetween the vectors p ′ and p , given by ˆp ′ · ˆp .Since Eqs. (2.3) and (2.5) are completely general, any arbitrary NN force can be cast intothis form and serve as input. Finalizing the formulation, we only need to antisymmetrizedin the initial state by applying (1 − P ) | p i| m m i| tm t i , and consider the on-shell t-matrixelement for given tm t : M tm t m ′ m ′ ,m m ≡ − m π ) t tm t m ′ m ′ ,m m ( p ′ , p ) (cid:12)(cid:12)(cid:12) on − shell = − m π ) (cid:16) h m ′ m ′ | (cid:2) t tm t ( p ′ , p ) + ( − ) t t tm t ( p ′ , − p ) P s (cid:3) | m m i (cid:17) . (2.9)Here P s interchanges the spin magnetic quantum numbers for the initial particles, m rep-resents the nucleon mass.For the on-shell condition, characterized by | p ′ | = | p | the vectors p − p ′ and p + p ′ are orthogonal. Under this condition, the operator σ · σ can be represented as a linearcombination of the operators w j , j = 4 − σ · σ = 1( p × p ′ ) σ · ( p × p ′ ) σ · ( p × p ′ )+ 1( p + p ′ ) σ · ( p + p ′ ) σ · ( p + p ′ )+ 1( p − p ′ ) σ · ( p − p ′ ) σ · ( p − p ′ ) (2.10)We can use the relation of Eq. (2.10) for internal consistency checks of the calculations.However, in order to keep the most general off-shell structure of Eq. (2.5), we need to keepall six terms. We will come back to the numerical implications of this fact below.From Eq. (2.9) we read off that the scattering matrix is given by M tm t m ′ m ′ ,m m = − m π ) X j =1 (cid:20) t tm t j ( p ′ , p ) h m ′ m ′ | w j ( σ , σ , p ′ , p ) | m m i + ( − ) t t tm t j ( p ′ , − p ) h m ′ m ′ | w j ( σ , σ , p ′ , − p ) | m m i (cid:21) (2.11)On the other hand the standard form of the on-shell t-matrix for given quantum numbers tm t [18] reads in the Wolfenstein representation M tm t m ′ m ′ ,m m = a tm t h m ′ m ′ | m m i− i c tm t | p × p ′ | h m ′ m ′ | w ( σ , σ , p ′ , p ) | m m i + m tm t | p × p ′ | h m ′ m ′ | w ( σ , σ , p ′ , p ) | m m i + ( g + h ) tm t ( p + p ′ ) h m ′ m ′ | w ( σ , σ , p ′ , p ) | m m i + ( g − h ) tm t ( p − p ′ ) h m ′ m ′ | w ( σ , σ , p ′ , p ) | m m i (2.12)4ue to the action of P s in Eq. (2.9), which interchanges m with m , the two partsof Eq. (2.11) yield different results. Again, standard relations [18, 19] must be applied toextract the Wolfenstein parameters: a tm t = 14 Tr ( M ) c tm t = − i
18 Tr (cid:18)
M w ( σ , σ , p ′ , p ) | p × p ′ | (cid:19) m tm t = 14 Tr (cid:18) M w ( σ , σ , p ′ , p ) | p × p ′ | (cid:19) ( g + h ) tm t = 14 Tr (cid:18) M w ( σ , σ , p ′ , p )( p + p ′ ) (cid:19) ( g − h ) tm t = 14 Tr (cid:18) M w ( σ , σ , p ′ , p )( p − p ′ ) (cid:19) (2.13)It is straightforward to work out those relations starting from Eq. (2.11). In order to simplifythe notation we write t j ≡ t tm t j ( p ′ , p ), ˜ t j ≡ t tm t j ( p ′ , − p ), and x ≡ ˆp ′ · ˆp and obtain a tm t = t + ( − ) t h
12 ˜ t + 32 ˜ t + 12 p (1 − x )˜ t + p (1 − x )˜ t + p (1 + x )˜ t i c tm t = ip √ − x (cid:0) t − ( − ) t ˜ t (cid:1) m tm t = t + p (1 − x ) t + ( − ) t h
12 ˜ t −
12 ˜ t + 12 p (1 − x )˜ t − p (1 − x )˜ t − p (1 + x )˜ t i g tm t = t + p (1 + x ) t + p (1 − x ) t + ( − ) t h
12 ˜ t −
12 ˜ t − p (1 − x )˜ t i h tm t = p (1 + x ) t − p (1 − x ) t + ( − ) t h − p (1 − x )˜ t + p (1 + x )˜ t i (2.14)It remains to consider the particle representation. For the proton-proton or neutron-neutronsystem the isospin is t = 1. Thus the above given Wolfenstein parameters are already thephysical ones and enter the calculation of observables. In the case of the neutron-protonsystem both isospins contribute and the physical amplitudes are given by ( a + a ), ( c + c ), etc.Once the Wolfenstein parameters are known, all NN observables can readily be calculatedtaking well defined bilinear products thereof [18]. For example, the spin averaged differentialcross section I is given as Tr M M † .For completeness, we also give the derivation of the deuteron which carries isospin t = 0and total spin s = 1. We employ the operator form from Ref. [5], h p | Ψ m d i = (cid:20) φ ( p ) + (cid:18) σ · p σ · p − p (cid:19) φ ( p ) (cid:21) | m d i≡ X k =1 φ k ( p ) b k ( σ , σ , p ) | m d i , (2.15)where | m d i describes the state in which the two spin- states are coupled to the total spin-1and the magnetic quantum number m d . The definition of the operators b k can be easily readoff the first line of Eq. (2.15). The two scalar functions φ ( p ) and φ ( p ) are related in a5imple way to the standard s - and d -wave components of the deuteron wave function, ψ ( p )and ψ ( p ) by [5] ψ ( p ) = φ ( p ) ,ψ ( p ) = 4 p √ φ ( p ) . (2.16)Next we use the Schr¨odinger equation in integral form projected on isospin states,Ψ m d = G V Ψ m d . (2.17)Inserting the explicit expression of Eq. (2.15) we obtain h φ ( p ) + (cid:18) σ · p σ · p − p (cid:19) φ ( p ) i | m d i =1 E d − p m Z d p ′ X j =1 v j ( p , p ′ ) w j ( σ , σ , p , p ′ ) × h φ ( p ′ ) + (cid:18) σ · p ′ σ · p ′ − p ′ (cid:19) φ ( p ′ ) i | m d i , (2.18)where E d is the deuteron binding energy. We remove the spin dependence by projectingfrom the left with h m d | b k ( σ , σ , p ) and summing over m d . This leads to X m d = − h m d | b k ( σ , σ , p ) X k ′ =1 φ k ′ ( p ) b k ′ ( σ , σ , p ) | m d i =1 E d − p m X m d = − Z d p ′ X j =1 v j ( p , p ′ ) w j ( σ , σ , p , p ′ ) × X k ′′ =1 φ k ′′ ( p ′ ) b k ′′ ( σ , σ , p ′ ) | m d i . (2.19)Defining the scalar functions A dkk ′ ( p ) ≡ X m d = − h m d | b k ( σ , σ , p ) b k ′ ( σ , σ , p ) | m d i (2.20)and B dkjk ′′ ( p , p ′ ) ≡ X m d = − h m d | b k ( σ , σ , p ) w j ( σ , σ , p , p ′ ) b k ′′ ( σ , σ , p ′ ) | m d i , (2.21)we obtain for Eq. (2.19) X k ′ =1 A dkk ′ ( p ) φ k ′ ( p ) = 1 E d − p m Z d p ′ X j =1 v j ( p , p ′ ) X k ′′ =1 B dkjk ′′ ( p , p ′ ) φ k ′′ ( p ′ ) . (2.22)6ote that A dkk ′ and B dkjk ′′ are both independent of the interaction. Therefore, these coeffi-cients can be prepared beforehand for all calculations of the deuteron bound state, whichconsists of two coupled equations for the functions φ ( p ) and φ ( p ). The summation over m d guarantees the scalar nature of the functions A dkk ′ ( p ) and B dkjk ′′ ( p , p ′ ), which are given inAppendix B. The azimuthal angle can be trivially integrated out, leading to the final formof the deuteron equation X k ′ =1 A dkk ′ ( p ) φ k ′ ( p ) =2 πE d − p m X k ′′ =1 Z ∞ dp ′ p ′ φ k ′′ ( p ′ ) Z − dx X j =1 v j ( p, p ′ , x ) B dkjk ′′ ( p, p ′ , x ) , (2.23)where x ≡ ˆp ′ · ˆp . III. NUMERICAL REALIZATIONA. The deuteron
For a numerical treatment of Eq. (2.23), it is convenient to first define Z k,k ′ ( p, p ′ ) ≡ Z − dx X j =1 v j ( p, p ′ , x ) B dkjk ′ ( p, p ′ , x ) (3.1)and then assume that the integral over p ′ will be carried out with some choice of Gaussianpoints and weights ( p j , g j ) with j = 1 , , . . . , N . This leads to X k ′ =1 N X j =1 (cid:18) g j p j Z kk ′ ( p i , p j ) + δ ij p j mπ A dkk ′ ( p i ) (cid:19) φ k ′ ( p j ) = E d X k ′ =1 π A dkk ′ ( p i ) φ k ′ ( p i ) . (3.2)Eq. (3.2) can be written as a so-called generalized eigenvalue problem Rξ = E d Y ξ, (3.3)or N X l ′ =1 R ll ′ ξ l ′ = E d N X l ′ =1 Y ll ′ ξ l ′ , (3.4)where l = i + ( k − N,ξ l ′ = φ k ′ ( p j ) , l ′ = j + ( k ′ − N,R ll ′ = g j p j Z kk ′ ( p i , p j ) + δ ij p j mπ A dkk ′ ( p i ) Y ll ′ = δ ij π A dkk ′ ( p i ) . (3.5)7 ABLE I: The parameters of the chiral potential of Ref. [13] in order NNLO. The LEC’s are givenfor the cutoff combination Λ= 600 MeV and ˜Λ= 700 MeV. The pion decay constant F π and massesare given in MeV. The constants c i are given in GeV − , C S and C T in GeV − and the other C i inGeV − g A F π m π m π ± m c c c C S C T C C C C C C C -112.932 2.60161 385.633 1343.49 -121.543 -614.322 1269.04 -26.4880 -1385.12TABLE II: Meson parameters for the Bonn B potential [15]. The σ parameters shown in the tableare for NN total isospin 0. For NN total isospin 1 they should be replaced by m σ = 550 MeV, g α π = 8 . α = 1 . n = 1.meson m α [MeV] g α π f α g α Λ α [GeV] n π η δ
983 2.488 2 1 σ
720 18.3773 2 1 ρ
769 0.9 6.1 1.85 2 ω Since A d = 0, A d = A d = 0 and A d = 0, the matrix Y is diagonal and can be easilyinverted, we encounter an eigenvalue problem (cid:0) Y − R (cid:1) ξ = E d ξ, (3.6)which is of the same type and dimension as is being solved for the deuteron wave function ina standard partial wave representation, where one calculates the s - and d -wave components, ψ ( p ) and ψ ( p ). The connection between the two solutions, ( φ ( p ) , φ ( p )) and ( ψ ( p ) , ψ ( p )),given by Eqs. (2.16) provides a direct check of the numerical accuracy.As a first example we use a chiral NNLO potential [13], which for the convenience ofthe reader is briefly described in Appendix C. For the specific calculation performed herewe take the neutron-proton version of this potential and employ the parameters listed inTable I.We consistently use these potential parameters in the 3D and the PW calculations. Inthe first case we solve Eq. (3.6) for φ ( p ) and φ ( p ) and then use Eqs. (2.16) to obtain ψ ( p )and ψ ( p ). In the second case we employ the standard partial wave representation of thepotential and solve the Schr¨odinger equation directly for ψ ( p ) and ψ ( p ). Both methodsgive the same value for the deuteron binding energy, namely E d =-2.19993 MeV and s -stateprobability P s =95.291 %. The wave functions are identical as can be seen in Fig. 1.As second NN force we choose the Bonn B potential [15], which has a more intricatestructure due to the different meson-exchanges and the Dirac spinors. The operator formof this potential, corresponding to the basis of Eq. (2.2) is derived in Appendix D and theparameters are given in Table II. In this case the nucleon mass is set to m = 939.039 MeV.8gain we have an excellent agreement between the 3D and the partial wave based calculationfor the deuteron binding energy, E d =-2.2242 MeV, the s -state probability ( P s = 95.014 %)and the wave functions, which are displayed in Fig. 2.In summary, we confirm that the 3D approach gives numerically stable results, which arein perfect agreement with the calculations based on standard partial wave methods. ψ ( p ) [f m / ] p [fm -1 ] -0.25-0.2-0.15-0.1-0.05 0 0 1 2 3 4 ψ ( p ) [f m / ] p [fm -1 ] FIG. 1: The s -wave (left) and d -wave (right) component of the deuteron wave function as a functionof the relative momentum p for the chiral NNLO potential specified in the text. Crosses representresults obtained with the operator approach and solid lines are from the standard partial wavedecomposition. B. NN scattering observables
The inhomogeneous LS equation (2.6) for the six components t tm t j can be solved for afixed value of p . For the vectors ˆp and ˆp ′ we choose the explicit representation ˆp = (0 , , ˆp ′ = ( p − x ′ , , x ′ ) ˆp ′′ = ( p − x ′′ cos ϕ ′′ , p − x ′′ sin ϕ ′′ , x ′′ ) (3.7)so that the scalar products become ˆp ′ · ˆp = x ′ ˆp ′′ · ˆp = x ′′ ˆp ′ · ˆp ′′ = x ′ x ′′ + p − x ′ p − x ′′ cos ϕ ′′ ≡ y. (3.8)Let us now calculate the integral term on the right-hand-side of Eq. (2.6) for a positiveenergy of the NN system, E c.m. ≡ p m : S k ( p ′ , p, x ′ ) ≡ ¯ p Z dp ′′ p ′′ p − p ′′ + iǫ f k ( p ′′ ; p ′ , p, x ′ ) , (3.9)9 ψ ( p ) [f m / ] p [fm -1 ] -0.25-0.2-0.15-0.1-0.05 0 0 1 2 3 4 ψ ( p ) [f m / ] p [fm -1 ] FIG. 2: The same as in Fig. 1 but for the Bonn B potential [15]. where f k ( p ′′ ; p ′ , p, x ′ ) ≡ f k ( p ′′ ) ≡ m X j,j ′ =1 1 Z − dx ′′ π Z dϕ ′′ B kjj ′ ( p ′ , p ′′ , p, x ′ , x ′′ , ϕ ′′ ) v j ( p ′ , p ′′ , y ) t j ′ ( p ′′ , p, x ′′ ) . (3.10)Here the index tm t for the t-matrix element is omitted for simplicity. For the momentumintegration in Eq. (3.9) an upper bound ¯ p is introduced, since the contributions to theintegral for larger momenta are insignificant, the potential and the t -matrix are essentiallyzero. Then the integral of Eq. (3.9) can be treated in a standard fashion and one obtains S k ( p ′ , p, x ′ ) = ¯ p Z dp ′′ p ′′ f k ( p ′′ ) − p f k ( p ) p − p ′′ + 12 p f k ( p ) (cid:18) ln ¯ p + p ¯ p − p − iπ (cid:19) . (3.11)It is tempting to solve Eq. (2.6) by iteration and then sum the resulting Neumann serieswith a Pad´e scheme. The determinant of the 6 × A ( p ′ , p, x ′ ), which appears onboth sides of (2.6), can be easily calculated with the resultdet( A ) = − p p ′ (cid:16) p − p ′ (cid:17) (cid:16) − x ′ (cid:17) . (3.12)In particular, this determinant is zero for p ′ = p and x ′ = ±
1. However, by a careful choiceof the p , p ′ and x ′ points, it is possible to work with non-zero values of det( A ), so that thematrix A can be inverted. In this case Eq. (2.6) can be written as t ( p ′ , p, x ′ ) = v ( p ′ , p, x ′ ) + A − ( p ′ , p, x ′ ) S ( p ′ , p, x ′ ) , (3.13)where t ( p ′ , p, x ′ ), v ( p ′ , p, x ′ ) and S ( p ′ , p, x ′ ) denote now six-dimensional vectors with compo-nents t j , v j and S j . Note that S ( p ′ , p, x ′ ) contains the unknown vector t ( p ′ , p, x ′ ). We arriveat the following iteration scheme: t (1) ( p ′ , p, x ′ ) = v ( p ′ , p, x ′ ) t ( n ) ( p ′ , p, x ′ ) = v ( p ′ , p, x ′ ) + A − ( p ′ , p, x ′ ) S ( n − ( p ′ , p, x ′ ) , for n > , (3.14)10here S ( n − ( p ′ , p, x ′ ) is calculated using the vector t ( p ′ , p, x ′ ) from the previous iteration,i.e. t ( n − ( p ′ , p, x ′ ). However, our experience with this iteration scheme is discouraging.Numerically det( A ) can be very close to zero, and in such cases the rank of matrix A canvary from 2 to 5. As a consequence, it is very difficult to maintain numerical stability forthis iterative method. Another drawback of using the inverse of A is that it is impossible toobtain the on-shell matrix element t ( p , p , x ′ ) directly. One would have to rely on numericalinterpolations for calculating on-shell matrix elements.For this reason we decided to solve Eq. (2.6) directly as a system of inhomogeneouscoupled algebraic equations. To this aim we first perform a discretization with respect tothe different variables in the problem. As typical grid sizes we take n x = 36 Gaussian pointsfor the x ′′ integration, and use the same grid for the x ′ points. Furthermore, we use n p = 36Gaussian points for the p ′ and p ′′ grids, which are defined the interval (0 , ¯ p = 40 fm − ).These points are distributed in such a way that p is avoided and the same number of pointsis put symmetrically into two narrow intervals on each side of p [16]. Such a choice provedadvantageous in the treatment of the S channel for the PWD calculations and is kept here.In addition, p is added to the set of p ′ points. Finally, we choose n ϕ ′′ = 60 Gaussian pointsfor the ϕ ′′ integration. Thus, we arrive at a system of 6 × ( n p + 1) × n x linear equations ofthe form Hξ = b, (3.15)where the vector ξ represents all unknown values of t j ( p ′ , p, x ′ ) for fixed p . If we choose fromthe very beginning p = p , then the solution of Eq. (3.15) contains the on-shell t-matrix inthe operator form, namely t j ( p , p , x ′ ).It is clear that for the on-shell t -matrix the solution cannot be unique, since the six oper-ators become linearly dependent on each other (see Eq. (2.10)). In principle, one thereforeexpects that Eq. (3.15) is non-invertible and that tools like a singular value decompositionare required for the solution. However, we found that this is not required since the stan-dard LU decomposition of Numerical Recipes [20] worked safely for both interactions, allthe considered laboratory energies and different choices of the mesh points. Interestingly,the actual solution for the on-shell t -matrix is not unique as expected and depends even onthe optimization level of the compiler. However, the observables turn out to be stable andunique.Of course, setting p = p is not necessary. For p = p the system of equations (3.15) hasa unique and smooth solution and afterwards the interpolation to the on shell case can besafely performed.The path to NN observables is straightforward. From Eq. (2.11) we evaluate first thescattering matrix M for all possible spin projections m ′ , m ′ , m , and m , noting thaton-shell t tm t j ( p ′ , p ) = t tm t j ( p , p , x ′ ) (3.16)and t tm t j ( p ′ , − p ) = t tm t j ( p , p , − x ′ ) . (3.17)Since we use a set of x ′ points which is symmetric with respect to x ′ = 0, no interpolationis required and M is easily obtained. Before we can make use of Eq. (2.13), we calculate11atrix elements of the modified operators w j appearing in (2.13), in the same representationas for the matrix M : D m ′ m ′ (cid:12)(cid:12)(cid:12) w ( σ , σ , p ′ , p ) | p × p ′ | (cid:12)(cid:12)(cid:12) m m ED m ′ m ′ (cid:12)(cid:12)(cid:12) w ( σ , σ , p ′ , p ) | p × p ′ | (cid:12)(cid:12)(cid:12) m m ED m ′ m ′ (cid:12)(cid:12)(cid:12) w ( σ , σ , p ′ , p )( p + p ′ ) (cid:12)(cid:12)(cid:12) m m ED m ′ m ′ (cid:12)(cid:12)(cid:12) w ( σ , σ , p ′ , p )( p − p ′ ) (cid:12)(cid:12)(cid:12) m m E . (3.18)For this calculation symbolic software like Mathematica c (cid:13) [21] proves very useful. In thenext step, the Wolfenstein parameters are calculated as sums over m ′ , m ′ , m and m . Forexample a tm t = 14 X m ′ ,m ′ X m ,m M tm t m ′ m ′ ,m m δ m ′ m δ m ′ m ,c tm t = − i X m ′ ,m ′ X m ,m M tm t m ′ m ′ ,m m D m m (cid:12)(cid:12)(cid:12) w ( σ , σ , p ′ , p ) | p × p ′ | (cid:12)(cid:12)(cid:12) m ′ m ′ E . (3.19)Finally, the NN observables result from the Wolfenstein parameters as simple bilinear ex-pressions [18].In Figs. 3–6 we compare a selected set of observables calculated with the new 3D methodto results obtained by using a standard partial wave decomposition, employing the samepotentials we used for the deuteron calculations. For the chiral potential we chose twolaboratory kinetic energies 13 and 150 MeV, whereas for the Bonn B potential the higherenergy is chosen to be 300 MeV. We made sure that in all cases a sufficient number of partialwaves is included to obtain converged results in the standard PWD approach. For all theenergies considered our converged PWD results agree perfectly with predictions obtainedfrom the new 3D approach.In Figs. 7–8 we demonstrate the convergence with respect to different maximum totalangular momenta j max towards the results calculated using our new 3D method for thedifferential cross section and the asymmetry A . Here we employ the Bonn B potential andshow the calculations for the neutron-neutron and neutron-proton cases separately. As onecan see, quite a sizeable number of partial waves is required for a converged calculationat 300 MeV. Finally, in Fig. 9 we display the Wolfenstein amplitudes for neutron-protonscattering at 300 MeV laboratory kinetic energy. Again we compare partial wave basedcalculations for different maximum total angular momenta j max to the 3D calculation. Weobserve that the maximum number of partial waves needed for obtaining a converged resultis quite different for the different amplitudes. IV. SUMMARY AND CONCLUSIONS
Two nucleon scattering at intermediate energies of a few hundred MeV requires quitea few angular momentum states in order to achieve convergence of e.g. scattering observ-ables. We formulated and numerically illustrated an approach to treating the NN system12 σ [ m b / s r ]
57 58 59 60 61 62 63 0 20 40 60 80 100 120 140 160 180-0.1-0.09-0.08-0.07-0.06-0.05-0.04-0.03-0.02-0.01 0 0 20 40 60 80 100 120 140 160 180 R A -0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 0 20 40 60 80 100 120 140 160 180-0.05-0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0 20 40 60 80 100 120 140 160 180 D Θ c.m. [deg] 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0 20 40 60 80 100 120 140 160 180 Θ c.m. [deg] FIG. 3: Selected observables for the neutron-neutron (left panel) and neutron-proton (right panel)system at the projectile laboratory kinetic energy 13 MeV as a function of the center of mass angle θ for the chiral NNLO potential [13]. Crosses represent results obtained with the operator approachand solid lines represent fully converged results from the standard PWD. For the definition of the R , A and D observables see e.g. [18]. σ [ m b / s r ]
57 58 59 60 61 62 63 64 0 20 40 60 80 100 120 140 160 180-0.1-0.09-0.08-0.07-0.06-0.05-0.04-0.03-0.02-0.01 0 0 20 40 60 80 100 120 140 160 180 R A -0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 0 20 40 60 80 100 120 140 160 180-0.06-0.05-0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0 20 40 60 80 100 120 140 160 180 D Θ c.m. [deg] 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0 20 40 60 80 100 120 140 160 180 Θ c.m. [deg] FIG. 4: The same as in Fig. 3 for the Bonn B potential [15]. σ [ m b / s r ] R -0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0 20 40 60 80 100 120 140 160 180-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0 20 40 60 80 100 120 140 160 180 A -0.6-0.4-0.2 0 0.2 0.4 0.6 0 20 40 60 80 100 120 140 160 180-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0 20 40 60 80 100 120 140 160 180 D Θ c.m. [deg] -0.4-0.2 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140 160 180 Θ c.m. [deg] FIG. 5: The same as in Fig. 3 for the projectile laboratory kinetic energy being 150 MeV. σ [ m b / s r ] R -0.2-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 20 40 60 80 100 120 140 160 180-0.4-0.3-0.2-0.1 0 0.1 0.2 0 20 40 60 80 100 120 140 160 180 A -0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140 160 180-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 20 40 60 80 100 120 140 160 180 D Θ c.m. [deg] -0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140 160 180 Θ c.m. [deg] FIG. 6: The same as in Fig. 4 for the projectile laboratory kinetic energy being 300 MeV. σ [ m b / s r ] -0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0 20 40 60 80 100 120 140 160 180 A Θ c.m. [deg] j max = 3 j max = 6 j max = 9 j max =12 j max =15 j max =18 FIG. 7: The convergence of the PWD results for the differential cross section and the depolariza-tion coefficient A [18] for neutron-neutron scattering based on different numbers of partial wavesdetermined by the maximal total angular momentum j max of the NN system (lines) with respectto the result of the three-dimensional calculation (crosses) for projectile laboratory kinetic energy300 MeV and the Bonn B potential [15]. working directly with momentum vectors and using spin-momentum operators multipliedby scalar functions, which only depend on the momentum vectors. This approach is quitenatural, since any general NN force being invariant under time-reversal, parity and Galilei(or Lorentz) transformations can only depend on six linear independent spin-momentum op-erators. The representation of the NN potential using spin-momentum operators leads to a17 σ [ m b / s r ] j max = 3 j max = 6 j max = 9 j max =12 j max =15 j max =18 -0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140 160 180 A Θ c.m. [deg] FIG. 8: The same as in Fig. 7 for neutron-proton scattering. system of six coupled equations of scalar functions (depending on momentum vectors) for theNN t-matrix, once the spin-momentum operators are analytically calculated by performingsuitable trace operations.We calculated deuteron properties and NN scattering observables using two different NNpotentials, one derived from chiral effective field theory and one from meson exchange. Forall cases we found perfect agreement between the calculations based on our new method andconventional calculations using a partial wave basis.This work is intended to serve as starting point towards treating three-nucleon systemswithout partial waves. The theoretical formulation has already been given for the 3N boundstate (including 3N forces) in [10] and for 3N scattering in [11]. For a much simpler case when18 R e ( m ) [f m ] j max = 15j max = 9j max = 6j max = 3 -0.050.000.050.10 I m ( m ) [f m ] -0.4-0.20.00.2 R e ( g ) [f m ] -0.050.000.050.10 I m ( g ) [f m ] Θ c.m. [deg]-0.4-0.20.0 R e ( h ) [f m ] Θ c.m. [deg] 0.000.040.08 I m ( h ) [f m ] -0.20.00.20.40.60.8 R e ( a ) [f m ] -0.20.00.20.4 I m ( a ) [f m ] -0.050.000.05 R e ( c ) [f m ] I m ( c ) [f m ] FIG. 9: The Wolfenstein parameters for neutron-proton scattering for projectile laboratory kineticenergy 300 MeV calculated with the Bonn B potential [15]. Results of the 3D calculation are givenby the crosses. The convergence of the PWD results for increasing values of maximum angularmomentum j max is shown by the different curves labeled in the figure. The left panels show thereal parts of the amplitudes, whereas the imaginary parts are displayed in the right panels. Acknowledgments
We thank Dr. Evgeny Epelbaum for providing us with a code for the operator form ofthe chiral NNLO potential.This work was supported by the Polish Ministry of Science and Higher Education un-der Grants No. N N202 104536 and No. N N202 077435 and in part under the auspicesof the U. S. Department of Energy, Office of Nuclear Physics under contract No. DE-FG02-93ER40756 with Ohio University. It was also partially supported by the HelmholtzAssociation through funds provided to the virtual institute “Spin and strong QCD”(VH-VI-231). The numerical calculations were partly performed on the supercomputer cluster ofthe JSC, J¨ulich, Germany.
Appendix A: Coefficients for NN scattering
In this appendix we present the expressions A and B given in Eqs. (2.7) and (2.8) forNN scattering. The coefficients A ij ( p ′ , p ) can be obtained in terms of the following fourfunctions F A F A ( p ′ , p ) = 4( p × p ′ ) (A1) F A ( p ′ , p ) = 4( p ′ + p ) (A2) F A ( p ′ , p ) = 4( p ′ − p ) (A3) F A ( p ′ , p ) = 4( p ′ − p ) (A4)20he non-zero coefficients A ij ( p ′ , p ) are: A ( p ′ , p ) = 4 (A5) A ( p ′ , p ) = 12 (A6) A ( p ′ , p ) = A ( p ′ , p ) = F A ( p ′ , p ) (A7) A ( p ′ , p ) = A ( p ′ , p ) = F A ( p ′ , p ) (A8) A ( p ′ , p ) = A ( p ′ , p ) = F A ( p ′ , p ) (A9) A ( p ′ , p ) = − F A ( p ′ , p ) (A10) A ( p ′ , p ) = 14 A ( p ′ , p ) (A11) A ( p ′ , p ) = 14 A ( p ′ , p ) (A12) A ( p ′ , p ) = A ( p ′ , p ) = F A ( p ′ , p ) (A13) A ( p ′ , p ) = 14 A ( p ′ , p ) (A14)All other A ij ( p ′ , p ) = 0.The non-vanishing coefficients B ikj ( p ′ , p ′′ , p ) can be expressed by means of the following25 functions F B : F B a ( p ′ , p ′′ , p ) = 4( p × p ′′ ) (A15) F B b ( p ′ , p ′′ , p ) = 4( p ′′ × p ′ ) (A16) F B c ( p ′ , p ′′ , p ) = 4( p × p ′ ) (A17) F B a ( p ′ , p ′′ , p ) = 4( p ′′ + p ) (A18) F B b ( p ′ , p ′′ , p ) = 4( p ′ + p ′′ ) (A19) F B c ( p ′ , p ′′ , p ) = 4( p ′ + p ) (A20) F B a ( p ′ , p ′′ , p ) = 4( p ′′ − p ) (A21) F B b ( p ′ , p ′′ , p ) = 4( p ′ − p ′′ ) (A22) F B c ( p ′ , p ′′ , p ) = 4( p ′ − p ) (A23) F B a ( p ′ , p ′′ , p ) = − p ′′ × p ′ ) · ( p × p ′′ ) (A24) F B b ( p ′ , p ′′ , p ) = − p × p ′ ) · ( p ′′ × p ′ ) (A25) F B c ( p ′ , p ′′ , p ) = − p × p ′ ) · ( p × p ′′ ) (A26) F B ( p ′ , p ′′ , p ) = 4 { ( p × p ′ ) · p ′′ } (A27) F B a ( p ′ , p ′′ , p ) = 2 { ( p ′ + p ′′ ) · ( p ′′ + p ) } (A28) F B b ( p ′ , p ′′ , p ) = 2 { ( p ′ + p ) · ( p ′ + p ′′ ) } (A29) F B c ( p ′ , p ′′ , p ) = 2 { ( p ′ + p ) · ( p ′′ + p ) } (A30)21 B a ( p ′ , p ′′ , p ) = 2 { ( p ′ − p ′′ ) · ( p ′′ − p ) } (A31) F B b ( p ′ , p ′′ , p ) = 2 { ( p ′ − p ) · ( p ′ − p ′′ ) } (A32) F B c ( p ′ , p ′′ , p ) = 2 { ( p ′ − p ) · ( p ′′ − p ) } (A33) F B a ( p ′ , p ′′ , p ) = 2 { ( p ′ + p ′′ ) · ( p ′′ − p ) } (A34) F B b ( p ′ , p ′′ , p ) = 2 { ( p ′ + p ) · ( p ′ − p ′′ ) } (A35) F B c ( p ′ , p ′′ , p ) = 2 { ( p ′ − p ) · ( p ′′ + p ) } (A36) F B a ( p ′ , p ′′ , p ) = 2 { ( p ′ − p ′′ ) · ( p ′′ + p ) } (A37) F B b ( p ′ , p ′′ , p ) = 2 { ( p ′ − p ) · ( p ′ + p ′′ ) } (A38) F B c ( p ′ , p ′′ , p ) = 2 { ( p ′ + p ) · ( p ′′ − p ) } (A39)The non-zero B ikj ( p ′ , p ′′ , p ): B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = 12 (A40) B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = F B a ( p ′ , p ′′ , p ) (A41) B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = F B a ( p ′ , p ′′ , p ) (A42) B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = F B a ( p ′ , p ′′ , p ) (A43) B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = F B a ( p ′ , p ′′ , p ) (A44) B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = F B b ( p ′ , p ′′ , p ) (A45) B ( p ′ , p ′′ , p ) = 116 F B a ( p ′ , p ′′ , p ) (A46) B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p )= B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p )= B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p )= B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = F B ( p ′ , p ′′ , p ) (A47) B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = F B b ( p ′ , p ′′ , p ) (A48) B ( p ′ , p ′′ , p ) = F B a ( p ′ , p ′′ , p ) (A49) B ( p ′ , p ′′ , p ) = F B a ( p ′ , p ′′ , p ) (A50) B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = F B b ( p ′ , p ′′ , p ) (A51) B ( p ′ , p ′′ , p ) = F B a ( p ′ , p ′′ , p ) (A52) B ( p ′ , p ′′ , p ) = F B a ( p ′ , p ′′ , p ) (A53) B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = F B c ( p ′ , p ′′ , p ) (A54) B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = F B c ( p ′ , p ′′ , p ) (A55) B ( p ′ , p ′′ , p ) = 116 F B c ( p ′ , p ′′ , p ) (A56) B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = F B c ( p ′ , p ′′ , p ) (A57) B ( p ′ , p ′′ , p ) = F B c ( p ′ , p ′′ , p ) (A58) B ( p ′ , p ′′ , p ) = F B c ( p ′ , p ′′ , p ) (A59) B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = F B c ( p ′ , p ′′ , p ) (A60)22 ( p ′ , p ′′ , p ) = F B c ( p ′ , p ′′ , p ) (A61) B ( p ′ , p ′′ , p ) = F B c ( p ′ , p ′′ , p ) (A62) B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = F B b ( p ′ , p ′′ , p ) (A63) B ( p ′ , p ′′ , p ) = 116 F B b ( p ′ , p ′′ , p ) (A64) B ( p ′ , p ′′ , p ) = F B b ( p ′ , p ′′ , p ) (A65) B ( p ′ , p ′′ , p ) = F B b ( p ′ , p ′′ , p ) (A66) B ( p ′ , p ′′ , p ) = F B b ( p ′ , p ′′ , p ) (A67) B ( p ′ , p ′′ , p ) = F B b ( p ′ , p ′′ , p ) (A68) B ( p ′ , p ′′ , p ) = 4 (A69) B ( p ′ , p ′′ , p ) = − p ′′ F B ( p ′ , p ′′ , p ) (A70) B ( p ′ , p ′′ , p ) = − F B b ( p ′ , p ′′ , p ) F B a ( p ′ , p ′′ , p )+ F B ( p ′ , p ′′ , p ) (A71) B ( p ′ , p ′′ , p ) = − F B b ( p ′ , p ′′ , p ) F B a ( p ′ , p ′′ , p )+ F B ( p ′ , p ′′ , p ) (A72) B ( p ′ , p ′′ , p ) = − F B a ( p ′ , p ′′ , p ) F B b ( p ′ , p ′′ , p )+ F B ( p ′ , p ′′ , p ) (A73) B ( p ′ , p ′′ , p ) = − F B b ( p ′ , p ′′ , p ) F B a ( p ′ , p ′′ , p )+ F B a ( p ′ , p ′′ , p ) (A74) B ( p ′ , p ′′ , p ) = − F B b ( p ′ , p ′′ , p ) F B a ( p ′ , p ′′ , p )+ F B a ( p ′ , p ′′ , p ) (A75) B ( p ′ , p ′′ , p ) = − F B a ( p ′ , p ′′ , p ) F B b ( p ′ , p ′′ , p )+ F B ( p ′ , p ′′ , p ) (A76) B ( p ′ , p ′′ , p ) = − F B b ( p ′ , p ′′ , p ) F B a ( p ′ , p ′′ , p )+ F B a ( p ′ , p ′′ , p ) (A77) B ( p ′ , p ′′ , p ) = − F B b ( p ′ , p ′′ , p ) F B a ( p ′ , p ′′ , p )+ F B a ( p ′ , p ′′ , p ) (A78)23 ( p ′ , p ′′ , p ) = − p F B ( p ′ , p ′′ , p ) (A79) B ( p ′ , p ′′ , p ) = − F B c ( p ′ , p ′′ , p ) F B a ( p ′ , p ′′ , p )+ F B ( p ′ , p ′′ , p ) (A80) B ( p ′ , p ′′ , p ) = − F B c ( p ′ , p ′′ , p ) F B a ( p ′ , p ′′ , p )+ F B ( p ′ , p ′′ , p ) (A81) B ( p ′ , p ′′ , p ) = − F B a ( p ′ , p ′′ , p ) F B c ( p ′ , p ′′ , p )+ F B ( p ′ , p ′′ , p ) (A82) B ( p ′ , p ′′ , p ) = − F B a ( p ′ , p ′′ , p ) F B c ( p ′ , p ′′ , p )+ F B c ( p ′ , p ′′ , p ) (A83) B ( p ′ , p ′′ , p ) = − F B a ( p ′ , p ′′ , p ) F B c ( p ′ , p ′′ , p )+ F B c ( p ′ , p ′′ , p ) (A84) B ( p ′ , p ′′ , p ) = − F B a ( p ′ , p ′′ , p ) F B c ( p ′ , p ′′ , p )+ F B ( p ′ , p ′′ , p ) (A85) B ( p ′ , p ′′ , p ) = − F B a ( p ′ , p ′′ , p ) F B c ( p ′ , p ′′ , p )+ F B c ( p ′ , p ′′ , p ) (A86) B ( p ′ , p ′′ , p ) = − F B a ( p ′ , p ′′ , p ) F B c ( p ′ , p ′′ , p )+ F B c ( p ′ , p ′′ , p ) (A87) B ( p ′ , p ′′ , p ) = − p ′ F B ( p ′ , p ′′ , p ) (A88) B ( p ′ , p ′′ , p ) = − F B c ( p ′ , p ′′ , p ) F B b ( p ′ , p ′′ , p )+ F B ( p ′ , p ′′ , p ) (A89) B ( p ′ , p ′′ , p ) = − F B c ( p ′ , p ′′ , p ) F B b ( p ′ , p ′′ , p )+ F B ( p ′ , p ′′ , p ) (A90) B ( p ′ , p ′′ , p ) = − F B b ( p ′ , p ′′ , p ) F B c ( p ′ , p ′′ , p )+ F B ( p ′ , p ′′ , p ) (A91) B ( p ′ , p ′′ , p ) = − F B c ( p ′ , p ′′ , p ) F B b ( p ′ , p ′′ , p )+ F B b ( p ′ , p ′′ , p ) (A92) B ( p ′ , p ′′ , p ) = − F B c ( p ′ , p ′′ , p ) F B b ( p ′ , p ′′ , p )+ F B b ( p ′ , p ′′ , p ) (A93)24 ( p ′ , p ′′ , p ) = − F B b ( p ′ , p ′′ , p ) F B c ( p ′ , p ′′ , p )+ F B ( p ′ , p ′′ , p ) (A94) B ( p ′ , p ′′ , p ) = − F B c ( p ′ , p ′′ , p ) F B b ( p ′ , p ′′ , p )+ F B b ( p ′ , p ′′ , p ) (A95) B ( p ′ , p ′′ , p ) = − F B c ( p ′ , p ′′ , p ) F B b ( p ′ , p ′′ , p )+ F B b ( p ′ , p ′′ , p ) (A96) B ( p ′ , p ′′ , p ) = − F B a ( p ′ , p ′′ , p ) (A97) B ( p ′ , p ′′ , p ) = − F B a ( p ′ , p ′′ , p ) (A98) B ( p ′ , p ′′ , p ) = − F B a ( p ′ , p ′′ , p ) (A99) B ( p ′ , p ′′ , p ) = − F B b ( p ′ , p ′′ , p ) (A100) B ( p ′ , p ′′ , p ) = − F B b ( p ′ , p ′′ , p ) (A101) B ( p ′ , p ′′ , p ) = − F B b ( p ′ , p ′′ , p ) (A102) B ( p ′ , p ′′ , p ) = − F B c ( p ′ , p ′′ , p ) (A103) B ( p ′ , p ′′ , p ) = − F B c ( p ′ , p ′′ , p ) (A104) B ( p ′ , p ′′ , p ) = − F B c ( p ′ , p ′′ , p ) (A105) B ( p ′ , p ′′ , p ) = −
24 (A106) B ( p ′ , p ′′ , p ) = 14 F B a ( p ′ , p ′′ , p ) F B ( p ′ , p ′′ , p ) (A107) B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = − { ( p ′′ + p ) · p ′ } F B ( p ′ , p ′′ , p ) (A108) B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = − { ( p ′′ − p ) · p ′ } F B ( p ′ , p ′′ , p ) (A109) B ( p ′ , p ′′ , p ) = 12 { F B c ( p ′ , p ′′ , p ) − F B b ( p ′ , p ′′ , p ) − F B c ( p ′ , p ′′ , p ) } F B a ( p ′ , p ′′ , p ) (A110) B ( p ′ , p ′′ , p ) = − { F B c ( p ′ , p ′′ , p ) + F B b ( p ′ , p ′′ , p ) − F B c ( p ′ , p ′′ , p ) } F B a ( p ′ , p ′′ , p ) (A111) B ( p ′ , p ′′ , p ) = 12 { F B c ( p ′ , p ′′ , p ) − F B b ( p ′ , p ′′ , p )+ F B c ( p ′ , p ′′ , p ) } F B a ( p ′ , p ′′ , p ) (A112) B ( p ′ , p ′′ , p ) = − { F B c ( p ′ , p ′′ , p ) + F B b ( p ′ , p ′′ , p )+ F B c ( p ′ , p ′′ , p ) } F B a ( p ′ , p ′′ , p ) (A113) B ( p ′ , p ′′ , p ) = 14 F B c ( p ′ , p ′′ , p ) F B ( p ′ , p ′′ , p ) (A114)25 ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = 2 { ( p ′ + p ) · p ′′ } F B ( p ′ , p ′′ , p ) (A115) B ( p ′ , p ′′ , p ) = − B ( p ′ , p ′′ , p ) = 2 { ( p ′ − p ) · p ′′ } F B ( p ′ , p ′′ , p ) (A116) B ( p ′ , p ′′ , p ) = − { F B b ( p ′ , p ′′ , p ) + F B a ( p ′ , p ′′ , p ) − F B b ( p ′ , p ′′ , p ) } F B c ( p ′ , p ′′ , p ) (A117) B ( p ′ , p ′′ , p ) = − { F B b ( p ′ , p ′′ , p ) + F B a ( p ′ , p ′′ , p )+ F B b ( p ′ , p ′′ , p ) } F B c ( p ′ , p ′′ , p ) (A118) B ( p ′ , p ′′ , p ) = − { F B b ( p ′ , p ′′ , p ) − F B a ( p ′ , p ′′ , p ) − F B b ( p ′ , p ′′ , p ) } F B c ( p ′ , p ′′ , p ) (A119) B ( p ′ , p ′′ , p ) = − { F B b ( p ′ , p ′′ , p ) − F B a ( p ′ , p ′′ , p )+ F B b ( p ′ , p ′′ , p ) } F B c ( p ′ , p ′′ , p ) (A120) B ( p ′ , p ′′ , p ) = 14 F B b ( p ′ , p ′′ , p ) F B ( p ′ , p ′′ , p ) (A121) B ( p ′ , p ′′ , p ) = − B ( p ′ , p ′′ , p ) = 2 { ( p ′ + p ′′ ) · p } F B ( p ′ , p ′′ , p ) (A122) B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = − { ( p ′ − p ′′ ) · p } F B ( p ′ , p ′′ , p ) (A123) B ( p ′ , p ′′ , p ) = − { F B a ( p ′ , p ′′ , p ) − F B c ( p ′ , p ′′ , p )+ F B a ( p ′ , p ′′ , p ) } F B b ( p ′ , p ′′ , p ) (A124) B ( p ′ , p ′′ , p ) = 12 { F B a ( p ′ , p ′′ , p ) + F B c ( p ′ , p ′′ , p )+ F B a ( p ′ , p ′′ , p ) } F B b ( p ′ , p ′′ , p ) (A125) B ( p ′ , p ′′ , p ) = 12 { F B a ( p ′ , p ′′ , p ) − F B c ( p ′ , p ′′ , p ) − F B a ( p ′ , p ′′ , p ) } F B b ( p ′ , p ′′ , p ) (A126) B ( p ′ , p ′′ , p ) = − { F B a ( p ′ , p ′′ , p ) + F B c ( p ′ , p ′′ , p ) − F B a ( p ′ , p ′′ , p ) } F B b ( p ′ , p ′′ , p ) (A127) B ( p ′ , p ′′ , p ) = − F B a ( p ′ , p ′′ , p ) F B c ( p ′ , p ′′ , p ) (A128) B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p )= B ( p ′ , p ′′ , p ) = 2 F B ( p ′ , p ′′ , p ) (A129) B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p )= − F B ( p ′ , p ′′ , p ) (A130) B ( p ′ , p ′′ , p ) = − F B b ( p ′ , p ′′ , p ) F B a ( p ′ , p ′′ , p ) (A131) B ( p ′ , p ′′ , p ) = − F B c ( p ′ , p ′′ , p ) F B b ( p ′ , p ′′ , p ) (A132)26 ( p ′ , p ′′ , p ) = − { F B c ( p ′ , p ′′ , p ) − F B b ( p ′ , p ′′ , p ) − F B c ( p ′ , p ′′ , p ) } (A133) B ( p ′ , p ′′ , p ) = − { F B c ( p ′ , p ′′ , p ) + F B b ( p ′ , p ′′ , p ) − F B c ( p ′ , p ′′ , p ) } (A134) B ( p ′ , p ′′ , p ) = − { F B c ( p ′ , p ′′ , p ) − F B b ( p ′ , p ′′ , p )+ F B c ( p ′ , p ′′ , p ) } (A135) B ( p ′ , p ′′ , p ) = − { F B c ( p ′ , p ′′ , p ) + F B b ( p ′ , p ′′ , p )+ F B c ( p ′ , p ′′ , p ) } (A136) B ( p ′ , p ′′ , p ) = − { F B b ( p ′ , p ′′ , p ) + F B a ( p ′ , p ′′ , p ) − F B b ( p ′ , p ′′ , p ) } (A137) B ( p ′ , p ′′ , p ) = − { F B b ( p ′ , p ′′ , p ) + F B a ( p ′ , p ′′ , p )+ F B b ( p ′ , p ′′ , p ) } (A138) B ( p ′ , p ′′ , p ) = − { F B b ( p ′ , p ′′ , p ) − F B a ( p ′ , p ′′ , p ) − F B b ( p ′ , p ′′ , p ) } (A139) B ( p ′ , p ′′ , p ) = − { F B b ( p ′ , p ′′ , p ) − F B a ( p ′ , p ′′ , p )+ F B b ( p ′ , p ′′ , p ) } (A140) B ( p ′ , p ′′ , p ) = − { F B a ( p ′ , p ′′ , p ) − F B c ( p ′ , p ′′ , p )+ F B a ( p ′ , p ′′ , p ) } (A141) B ( p ′ , p ′′ , p ) = − { F B a ( p ′ , p ′′ , p ) + F B c ( p ′ , p ′′ , p )+ F B a ( p ′ , p ′′ , p ) } (A142) B ( p ′ , p ′′ , p ) = − { F B a ( p ′ , p ′′ , p ) − F B c ( p ′ , p ′′ , p ) − F B a ( p ′ , p ′′ , p ) } (A143) B ( p ′ , p ′′ , p ) = − { F B a ( p ′ , p ′′ , p ) + F B c ( p ′ , p ′′ , p ) − F B a ( p ′ , p ′′ , p ) } (A144) B ( p ′ , p ′′ , p ) = −{ ( p ′′ + p ) · p ′ } F B ( p ′ , p ′′ , p ) (A145) B ( p ′ , p ′′ , p ) = −{ ( p ′′ − p ) · p ′ } F B ( p ′ , p ′′ , p ) (A146) B ( p ′ , p ′′ , p ) = −{ ( p ′ + p ′′ ) · p } F B ( p ′ , p ′′ , p ) (A147) B ( p ′ , p ′′ , p ) = −{ ( p ′ − p ′′ ) · p } F B ( p ′ , p ′′ , p ) (A148)27 ( p ′ , p ′′ , p ) = −{ ( p ′ + p ) · p ′′ } F B ( p ′ , p ′′ , p ) (A149) B ( p ′ , p ′′ , p ) = −{ ( p ′ − p ) · p ′′ } F B ( p ′ , p ′′ , p ) (A150) B ( p ′ , p ′′ , p ) = − F B ( p ′ , p ′′ , p ) (A151) B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p ) = B ( p ′ , p ′′ , p )= B ( p ′ , p ′′ , p ) = − F B ( p ′ , p ′′ , p ) (A152) Appendix B: Coefficients for the deuteron
In this appendix we present the expressions A d and B d given in Eqs. (2.12) for thedeuteron. A d ( p ) = 3 A d ( p ) = A d ( p ) = 0 A d ( p ) = 83 p (B1)The coefficients B dkjj ′ ( p , p ′ ) are explicitly calculated as B d ( p , p ′ ) = B d ( p , p ′ ) = 3 B d ( p , p ′ ) = 0 B d ( p , p ′ ) = ( p × p ′ ) B d ( p , p ′ ) = ( p ′ + p ) B d ( p , p ′ ) = ( p ′ − p ) (B2) B d ( p , p ′ ) = B d ( p , p ′ ) = 4( p · p ′ ) − p p ′ B d ( p , p ′ ) = − p · p ′ ( p × p ′ ) B d ( p , p ′ ) = − p p ′ − p · p ′ ) + 569 p p ′ ( p · p ′ ) B d ( p , p ′ ) = 49 p p ′ ( p + p ′ ) − p p ′ ( p · p ′ ) −
43 ( p + p ′ )( p · p ′ ) B d ( p , p ′ ) = 49 p p ′ ( p + p ′ ) + 169 p p ′ ( p · p ′ ) −
43 ( p + p ′ )( p · p ′ ) (B3) B d ( p , p ′ ) = B d ( p , p ′ ) = B d ( p , p ′ ) = 0 B d ( p , p ′ ) = − p ( p ′ × p ) B d ( p , p ′ ) = 83 p + 163 p ( p · p ′ ) + 4( p · p ′ ) − p p ′ B d ( p , p ′ ) = 83 p − p ( p · p ′ ) + 4( p · p ′ ) − p p ′ (B4)The expressions for the functions B d k , k = 1 ,
6, can be obtained from the functions B d k byreplacing p ↔ p ′ . 28 ppendix C: Example of a chiral potential For this particular example we will use the next-to-next-to-leading order (NNLO) chiralpotential from Ref. [12].The leading-order (LO) NN potential in the two-nucleon center-of-mass system (CMS)reads [13]: V LO = − π ) g A F π σ · q σ · qq + M π τ · τ + C S (2 π ) + C T (2 π ) σ · σ , (C1)where q = p ′ − p , and m π , F π and g A denote the pion mass, the pion decay constant andthe nucleon axial coupling constants. At next-to-leading order (NLO) a renormalizationof the low energy constants (LECs) is required and the contribution from the Goldberger-Treiman discrepancy leads to a modified value of g A . The remaining contributions to theNN potential at this order are V NLO = − π ) τ · τ π F π L ˜Λ ( q ) h m π (5 g A − g A −
1) + q (23 g A − g A −
1) + 48 g A m π m π + q i − π ) g A π F π L ˜Λ ( q ) (cid:18) σ · q σ · q − σ · σ q (cid:19) + C (2 π ) q + C (2 π ) k + ( C (2 π ) q + C (2 π ) k ) σ · σ + C (2 π ) i σ + σ ) · q × k + C (2 π ) q · σ q · σ + C (2 π ) k · σ k · σ , (C2)where q ≡ | q | and k = ( p ′ + p ). The loop function L ˜Λ ( q ) is defined in the spectralfunction regularization (SFR) as [12] L ˜Λ ( q ) = θ ( ˜Λ − m π ) ω q ln ˜Λ ω + q s + 2 ˜Λ qωs m π ( ˜Λ + q ) , (C3)with the following abbreviations: ω = p m π + q and s = q ˜Λ − m π . Here, ˜Λ denotesthe ultraviolet cutoff in the mass spectrum of the two-pion-exchange potential.The contributions at NNLO again lead to the renormalization and/or redefinition of theLECs C S , C T , C , . . . C . The only new momentum dependence is due to the followingterms: V NNLO = − π ) g A πF π (cid:0) m π (2 c − c ) − c q (cid:1) (2 m π + q ) A ˜Λ ( q ) − π ) g A c πF π τ · τ (4 m π + q ) A ˜Λ ( q ) (cid:0) σ · q σ · q − q σ · σ (cid:1) , (C4)where c , c , c are new π N LECs and the loop function A ˜Λ ( q ) is given by A ˜Λ ( q ) = θ ( ˜Λ − m π ) 12 q arctan q ( ˜Λ − m π ) q + 2 ˜Λ m π . (C5)29he expressions of the potential given in Eqs. (C1), (C2), and (C4) show that this potentialcan be readily expressed in the operators w j , j = 1 , V ≡ V LO + V NLO + V NNLO . (C6)It requires regularization when inserted into the Lippmann-Schwinger equation, which isachieved by introducing a regulated potential of the form V reg ( p ′ , p ) ≡ e − ( p ′ / Λ ) V ( p ′ , p ) e − ( p / Λ ) , (C7)with the cut-off parameter Λ. Appendix D: Scalar functions for the Bonn B potential
For the convenience of the reader we give the expressions for the Bonn B potential fromRef. [15] in a form which is more suited for our three-dimensional calculations. The ex-pressions for the exchange of pseudo-scalar ( ps ), scalar ( s ), and vector ( v ) mesons are givenby V ps ( p ′ , p ) = g ps (2 π ) m r mE ′ r mE F ps [( p ′ − p ) ]( p ′ − p ) + m ps O ps W ′ WV s ( p ′ , p ) = g s (2 π ) m r mE ′ r mE F s [( p ′ − p ) ]( p ′ − p ) + m s O s W ′ WV v ( p ′ , p ) = 1(2 π ) m r mE ′ r mE F v [( p ′ − p ) ]( p ′ − p ) + m v ( g v O vv + 2 g v f v O vt + f v O tt ) W ′ W , (D1)where m α are the masses of the exchanged mesons, m the nucleon mass, E = p m + p and W = m + E . The crucial quantities are the operators O ps , O s , O vv , O vt and O tt , whichare given in terms of the Dirac spinors as O ps = 4 m W ′ W ¯ u ( p ′ ) γ u ( p )¯ u ( − p ′ ) γ u ( − p ) , (D2) O s = − m W ′ W ¯ u ( p ′ ) u ( p ) ¯ u ( − p ′ ) u ( − p ) , (D3) O vv = 4 m W ′ W ¯ u ( p ′ ) γ µ u ( p )¯ u ( − p ′ ) γ µ u ( − p ) , (D4) O vt = mW ′ W n m ¯ u ( p ′ ) γ µ u ( p )¯ u ( − p ′ ) γ µ u ( − p ) − ¯ u ( p ′ ) γ µ u ( p )¯ u ( − p ′ ) (cid:20) ( E ′ − E )( g µ − γ µ γ ) + ( p + p ′ ) µ (cid:21) u ( − p ) − ¯ u ( p ′ ) (cid:20) ( E ′ − E )( g µ − γ µ γ ) + ( p + p ′ ) µ (cid:21) u ( p )¯ u ( − p ′ ) γ µ u ( − p ) o , (D5) O tt = W ′ W n m ¯ u ( p ′ ) γ µ u ( p )¯ u ( − p ′ ) γ µ u ( − p ) − m ¯ u ( p ′ ) γ µ u ( p )¯ u ( − p ′ ) (cid:20) ( E ′ − E )( g µ − γ µ γ ) + ( p + p ′ ) µ (cid:21) u ( − p )30 m ¯ u ( p ′ ) (cid:20) ( E ′ − E )( g µ − γ µ γ ) + ( p + p ′ ) µ (cid:21) u ( p )¯ u ( − p ′ ) γ µ u ( − p )+ ¯ u ( p ′ ) (cid:20) ( E ′ − E )( g µ − γ µ γ ) + ( p + p ′ ) µ (cid:21) u ( p ) × ¯ u ( − p ′ ) (cid:20) ( E ′ − E )( g µ − γ µ γ ) + ( p + p ′ ) µ (cid:21) u ( − p ) o (D6)with ( p + p ′ ) µ = ( E + E ′ , p + p ′ ) and ( p + p ′ ) µ = ( E + E ′ , − p − p ′ ).These operators act in the spin spaces of nucleons 1 and 2: the bilinear forms built with¯ u ( p ′ ) . . . u ( p ) contain σ as acting in the spin space of nucleon 1 and the bilinear forms with¯ u ( − p ′ ) . . . u ( − p ) contain σ and act in the spin space of nucleon 2. The spinors u ( q ) arenormalized according to the definitions given in Ref. [23] and explicitly given as u ( q ) = r E + m m σ · q E + m ! . (D7)Each vertex is multiplied with a form factor F α [( p ′ − p ) ] = (cid:18) Λ α − m α Λ α + ( p ′ − p ) (cid:19) n . (D8)where the values of n and the cutoff parameters Λ α are given in Table II.Note that for the three iso-vector mesons ( π , δ and ρ ) contributing to the Bonn B poten-tial, expressions (D2)–(D6) are additionally multiplied by the isospin factor τ (1) · τ (2).In Ref. [5] this potential was presented in a different operator form. However, in that workone of the six operators was chosen to be σ · ( p + p ′ ) σ · ( p ′ − p ) + σ · ( p ′ − p ) σ · ( p + p ′ ) ,which is an operator that violates time reversal invariance. In practice, this operator isalways multiplied with the term ( p ′ − p ), which also violates time reversal invariance.Therefore, the entire term is invariant as it should be. In principle it is not desirable to workwith symmetry violating operators, thus we prefer to use the operators from Eq. (2.2) andrewrite σ · ( p + p ′ ) σ · ( p ′ − p ) + σ · ( p ′ − p ) σ · ( p + p ′ ) = − p × p ′ ) p ′ − p σ · σ + ( p − p ′ ) p ′ − p σ · ( p + p ′ ) σ · ( p + p ′ )+ ( p + p ′ ) p ′ − p σ · ( p − p ′ ) σ · ( p − p ′ ) + 4 p ′ − p σ · ( p × p ′ ) σ · ( p × p ′ ) , (D9)which is an identity for ( p + p ′ )( p ′ − p ) = p ′ − p = 0. Inserting Eq. (D9) into theexpressions given in [5] cancels the factor p ′ − p and one obtains the following expressionsfor the operators O α from Eqs. (D2)-(D6) in terms of the operators w j ≡ w j ( σ , σ , p ′ , p )from Eq. (2.2): O ps = (cid:18) mE ′ + E (cid:19) (cid:20) ( p ′ · p ) − p ′ p (cid:21) w + (cid:18) mE ′ + E (cid:19) w + 14 ( − ( W ′ − W ) + (cid:18) mE ′ + E (cid:19) [ p ′ + p − p ′ · p )] ) w
31 14 ( − ( W ′ + W ) + (cid:18) mE ′ + E (cid:19) [ p ′ + p + 2( p ′ · p )] ) w (D10) O s = − [ W ′ W − ( p ′ · p )] w − [ W ′ W − ( p ′ · p )] w + w (D11) O vv = (cid:8) [ W ′ W + ( p ′ · p )] + W ′ p + W p ′ + 2 W ′ W ( p ′ · p ) (cid:9) w + ( − (cid:0) W ′ + W (cid:1) (cid:0) p ′ + p (cid:1) + 2 W ′ W ( p ′ · p )+ 12 (cid:18) mE ′ + E (cid:19) (cid:2) p ′ + p − p ′ · p ) (cid:3)) w − [3 W ′ W + ( p ′ · p )] w − (cid:18) mE ′ + E (cid:19) w − (cid:26) − ( W ′ − W ) + (cid:18) mE ′ + E (cid:19) [ p ′ + p − p ′ · p )] (cid:27) w − (cid:26) − ( W ′ + W ) + (cid:18) mE ′ + E (cid:19) [ p ′ + p + 2( p ′ · p )] (cid:27) w (D12) O vt = (cid:26) W ′ p + W p ′ − W ′ − W m (cid:0) W ′ p − W p ′ (cid:1) + 2 W ′ W [2 W ′ W + ( p ′ · p )] − W ′ + Wm (cid:2) W ′ W − ( p ′ · p ) (cid:3)(cid:27) w + (cid:26) W ′ W ( p ′ · p ) − (cid:0) W ′ + W (cid:1) (cid:0) p ′ + p (cid:1) + 12 (cid:18) mE ′ + E (cid:19) (cid:2) p ′ + p − p ′ · p ) (cid:3) − m ( E ′ + E ) (cid:2) W ′ p + W p ′ − (cid:0) W ′ + W (cid:1) ( p ′ · p ) (cid:3)(cid:27) w − (cid:20) W ′ W + W ′ + Wm ( p ′ · p ) (cid:21) w + (cid:26) − W ′ + Wm + 12 m ( E ′ + E ) (cid:2) W ′ + W − m ( W ′ + W ) (cid:3)(cid:27) w − (cid:26) − ( W ′ − W ) + (cid:18) mE ′ + E (cid:19) [ p ′ + p − p ′ · p )] − m ( E ′ + E ) [ W ′ p + W p ′ − (cid:0) W ′ + W (cid:1) ( p ′ · p )] (cid:27) w − (cid:26) − ( W ′ + W ) + (cid:18) mE ′ + E (cid:19) [ p ′ + p + 2( p ′ · p )] − m ( E ′ + E ) [ W ′ p + W p ′ + (cid:0) W ′ + W (cid:1) ( p ′ · p )] (cid:27) w (D13) O tt = ( [ W ′ W + ( p ′ · p )] + 2 (cid:18) − W ′ + Wm (cid:19) (cid:2) W ′ W − ( p ′ · p ) (cid:3) (cid:26) W ′ W − m ( W ′ + W )] + ( p ′ · p )2 m (cid:27) [ W ′ W − ( p ′ · p )] + (cid:20) W ′ − W ) m (cid:21) (cid:0) W ′ p + W p ′ (cid:1) + 2 (cid:20) − ( W ′ − W ) m (cid:21) W ′ W ( p ′ · p ) − W ′ − Wm (cid:0) W ′ p − W p ′ (cid:1)) w + ( (cid:20) − ( W ′ − W ) m (cid:21) W ′ W ( p ′ · p ) − (cid:20) W ′ − W ) m (cid:21) (cid:18) mE ′ + E (cid:19) ( p ′ · p ) − (cid:20) W ′ − W ) m (cid:21) (cid:18) m + E ′ EE ′ + E + m (cid:19) ( W ′ p + W p ′ ) − m ( E ′ + E ) (cid:2) W ′ p + W p ′ − (cid:0) W ′ + W (cid:1) ( p ′ · p ) (cid:3)) w + ( − W ′ W − ( p ′ · p ) + 2 (cid:18) − W ′ + Wm (cid:19) ( p ′ · p ) − (cid:20) − ( W ′ − W ) m (cid:21) W ′ W + (cid:26) W ′ W − m ( W ′ + W )] + ( p ′ · p )2 m (cid:27) [ W ′ W − ( p ′ · p )] ) w − (cid:26) W ′ W − m ( W ′ + W )] + ( p ′ · p )2 m − (cid:18) − W ′ + Wm (cid:19) + (cid:20) W ′ − W ) m (cid:21) (cid:18) mE ′ + E (cid:19) − m ( E ′ + E ) (cid:0) W ′ + W (cid:1)) w − ((cid:20) − ( W ′ − W ) m (cid:21) W ′ W − m ( E ′ + E ) [ W ′ p + W p ′ − (cid:0) W ′ + W (cid:1) ( p ′ · p )]+ 12 (cid:20) W ′ − W ) m (cid:21) (cid:20)(cid:18) mE ′ + E (cid:19) [ p ′ + p − p ′ · p )] − W ′ − W (cid:21)) w − ( − (cid:20) − ( W ′ − W ) m (cid:21) W ′ W − m ( E ′ + E ) (cid:2) W ′ p + W p ′ + ( W ′ + W )( p ′ · p ) (cid:3) + 12 (cid:20) W ′ − W ) m (cid:21) (cid:20)(cid:18) mE ′ + E (cid:19) [ p ′ + p + 2( p ′ · p )] − W ′ − W (cid:21)) w . (D14)The values of the parameters are given in Table II. [1] W. Gl¨ockle, H. Wita la, D. H¨uber, H. Kamada, J. Golak, Phys. Rep. 274, 107 (1996).[2] A. Nogga, H. Kamada and W. Gl¨ockle, Phys. Rev. Lett. , 944 (2000).[3] Ch. Elster, W. Schadow, A. Nogga and W. Gl¨ockle, Few Body Syst. , 83 (1999).[4] H. Liu, Ch. Elster and W. Gl¨ockle, Phys. Rev. C , 054003 (2005).[5] I. Fachruddin, Ch. Elster, W. Gl¨ockle, Phys. Rev. C , 054003 (2001).[6] S. Bayegan, M. R. Hadizadeh and M. Harzchi, Phys. Rev. C , 064005 (2008).[7] G. Ramalho, A. Arriaga and M. T. Pena, Few Body Syst. , 123 (2006).[8] G. Caia, V. Pascalutsa and L. E. Wright, Phys. Rev. C , 034003 (2004).
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